PROC
Longtime dynamics for an elastic waveguide model
Zhijian Yang Ke Li
Conference Publications 2013, 2013(special): 797-806 doi: 10.3934/proc.2013.2013.797
The paper studies the longtime dynamics for a nonlinear wave equation arising in elastic waveguide model: $u_{tt}- \Delta u-\Delta u_{tt}+\Delta^2 u- \Delta u_t -\Delta g(u)=f(x)$. It proves that the equation possesses in trajectory phase space a global trajectory attractor $\mathcal{A}^{tr}$ and the full trajectory of the equation in $\mathcal{A}^{tr}$ is of backward regularity provided that the growth exponent of nonlinearity $g(u)$ is supercritical.
keywords: longtime dynamics global solution Nonlinear wave equation trajectory attractor backward regularity.
DCDS
Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models
Zhijian Yang Yanan Li
Discrete & Continuous Dynamical Systems - A 2018, 38(5): 2629-2653 doi: 10.3934/dcds.2018111

In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.

keywords: Pullback exponential attractor stability with respect to perturbations non-autonomous Kirchhoff wave equations structural damping supercritical nonlinearity
DCDS
Global attractor for a strongly damped wave equation with fully supercritical nonlinearities
Zhijian Yang Zhiming Liu
Discrete & Continuous Dynamical Systems - A 2017, 37(4): 2181-2205 doi: 10.3934/dcds.2017094

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: $ u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x) $. In the case when the nonlinearities $ h(u_t) $ and $ g(u) $ are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

keywords: Strongly damped wave equation subclass of limit solutions generalized semiflow supercritical nonlinearities global attractor
CPAA
Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$
Pengyan Ding Zhijian Yang
Communications on Pure & Applied Analysis 2019, 18(2): 825-843 doi: 10.3934/cpaa.2019040

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on $\mathbb{R}^N$: $u_{tt}-φ(x)Δ u_{t}-φ(x)M(\|\nabla u\|^{2})Δ u+f(u) = h(x)$. It proves that when the growth exponent $p$ of the nonlinearity $f(u) $ is up to the supercritical range: $ 1≤ p < p^{**}(\equiv \frac{N+4}{(N-4)^+})$, the related solution semigroup has in weighted energy space a (strong) global attractor and a partially strong exponential attractor, respectively. In particular, the partially strong exponential attractor becomes the strong one in non-supercritical case (i.e., $1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$).

keywords: Kirchhoff wave equation weighted energy space supercritical nonlinearity global attractor exponential attractor
DCDS
Longtime behavior of the semilinear wave equation with gentle dissipation
Zhijian Yang Zhiming Liu Na Feng
Discrete & Continuous Dynamical Systems - A 2016, 36(11): 6557-6580 doi: 10.3934/dcds.2016084
The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation: $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.
keywords: gentle dissipation exponential attractor. global attractor well-posedness Semilinear wave equation
CPAA
Attractors and their stability on Boussinesq type equations with gentle dissipation
Zhijian Yang Pengyan Ding Xiaobin Liu
Communications on Pure & Applied Analysis 2019, 18(2): 911-930 doi: 10.3934/cpaa.2019044

The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:$ u_{tt}+Δ^2 u+(-Δ)^{α} u_{t}-Δ f(u) = g(x)$, with $α∈ (0, 1)$. For general bounded domain $Ω\subset \mathbb{R}^N (N≥1)$, we show that there exists a critical exponent $p_α\equiv\frac{N+2(2α-1)}{(N-2)^+}$ depending on the dissipative index α such that when the growth p of the nonlinearity f(u) is up to the range: $1≤p <p_α$, (ⅰ) the weak solutions of the equations are of additionally global smoothness when $t>0$; (ⅱ) the related dynamical system possesses a global attractor $\mathcal{A}_α$ and an exponential attractor $\mathcal{A}^α_{exp}$ in natural energy space for each $α∈ (0, 1)$, respectively; (ⅲ) the family of global attractors $\{\mathcal{A}_α\}$ is upper semicontinuous at each point $α_0∈ (0,1] $, i.e., for any neighborhood U of $\mathcal{A}_{α_0}, \mathcal{A}_α\subset U$ when $|α-α_0|\ll 1$. These results extend those for structural damping case: $α∈ [1, 2)$ in [31,32].

keywords: Boussinesq type equations gentle dissipation well-posedness global attractor exponential attractor upper semicontinuity of attractors

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